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If a weighted Euler transformation is applied to the asymptotic series for ezE1(z) the remainder can be expressed as an integral. Examination of this integral shows that for a transformation of given order the smallest term of the resulting series remains at approximately a constant distance from the start of the series. If, however, there is no restriction on the order of transformation the remainder may be decreased to zero by increasing the number of terms used, but if z is close to the negative real axis the rate of decrease is small. A more general theorem for alternating real series and Taylor's series is also given.

Let T be a nonlinear Volterra integral operator of the form

(with I compact, a < b), whose kernel K = K(x, t, u) satisfies the following conditions:

with

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On reducing to zero the remainder of an asymptotic series

An approximation property of certain nonlinear Volterra integral operators

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142 results in 41Axx

On reducing to zero the remainder of an asymptotic series

An approximation property of certain nonlinear Volterra integral operators